Normal stresses
72
[Chap. 4
lO'r-----------------------, 01
6
6 6 6
0
6
I11*(W) I 11 (;)
66 6 66 6 III 6% 6
6
&
6
~
...
6
000
0
Q
0 0 0 0 0 0 0 0 0
I
I
10'
10
2
;/5-' or w /5-'
10
3
10
1
Fig. 4.14 The Cox-Merz rule applied to the polymer solution D1, which is a 2% w/v polyisobutylene (Oppanol B2(0) solution in dekalin. 25 0 C.
it is not difficult to deduce the exact relationships in the lower limits of frequency and shear rate: (4.23) -
'1'1 ( y) 2
(4.24) y---,O
The former relationship states that the viscosity measured in oscillatory shear in the zero-frequency limit is equal to the low shear viscosity measured in steady shear. Equation (4.24) is a relationship between the limiting values of dynamic rigidity and first normal stress difference. In many cases, it is easier to carry out dynamic measurements than steady shear measurements and (4.23) and (4.24) provide a means of estimating the levels of TI and '1'} (and hence N}) from measurements of TIl and G I. We note that in view of eqn. (4.23) and the fact that both TI and TIl are usually monotonic decreasing functions of y and w, respectively, various attempts have been made to develop empirical relationships between TI and TIl at other than the lower limits of shear rate and frequency. The most popular, and most successful in this respect, certainly for polymeric liquids, is the so-called Cox-Merz (1958) rule, which proposes that TI should be the same function of y as I TI* I is of w, where I TI* I is the modulus of the complex viscosity, i.e. (4.25)